creating delimiters and inequality signs

Question

I would like to have a delimiter whose upper half is the upper half of the delimiter [ and whose lower half is the lower half of the delimiter ( -- so that it be one continuous line, and also the matching delimiter whose upper half is the upper half of the delimiter ) and whose lower half is the lower half of the delimiter ] .

Correspondingly, I would like to have a binary (inequality) relation with the full-size sign > over the succession of a half-length = sign followed on the right by an empty space of the same horizontal extent as the half-length = sign, so that the the total horizontal extent of the combination of the the half-length = sign plus the empty space be the same as that of the full-size sign > . Such a binary relation could be called "greater than or half-equal to". Similarly, I would like to have a "less than or half-equal to" sign, which is a binary relation with the full-size sign < over a half-length = sign preceded on the left by an empty space of the same horizontal extent as the half-length = sign.

Can this be done without METAFONT (which I don't know at all)? Thank you very much for your help.

Answer 1

The “asymmetric” square brackets are \lfloor, \rfloor, \lceil and \rceil.

For the strange inequalities, here is some code:

\documentclass{article}
\usepackage{trimclip}

% the left half of =
\newcommand{\lhalfeq}[1]{\clipbox{0pt 0pt {.5\width} 0pt}{$#1=$}}
% the right half of =
\newcommand{\rhalfeq}[1]{\clipbox{{.5\width} 0pt 0pt 0pt}{$#1=$}}
% greater than or half-equal to
\newcommand{\gtheq}{\mathrel{\mathpalette\xgtheq\relax}}
\newcommand{\xgtheq}[2]{%
  \vcenter{\hbox{%
    \oalign{$#1>$\cr\lhalfeq{#1}\hidewidth\cr}%
  }}
}
% less than or half-equal to
\newcommand{\ltheq}{\mathrel{\mathpalette\xltheq\relax}}
\newcommand{\xltheq}[2]{%
  \vcenter{\hbox{%
    \oalign{$#1<$\cr\hidewidth\rhalfeq{#1}\cr}%
  }}
}

\begin{document}
$a\gtheq b\ltheq c$
\end{document}

---

Thanks to Heiko, a slightly better implementation for vertically centering the symbol.

\documentclass{article}
\usepackage{trimclip}

% the left half of =
\makeatletter
\newcommand{\lhalfeq}[1]{\clipbox{0pt 0pt {.5\width} 0pt}{\adjeq{#1}}}
% the right half of =
\newcommand{\rhalfeq}[1]{\clipbox{{.5\width} 0pt 0pt 0pt}{\adjeq{#1}}}

\newcommand{\adjeq}[1]{%
  \sbox0{$#1\vcenter{}$}%
  \raisebox{\dimexpr\height-2\ht0\relax}[2\dimexpr\height-\ht0\relax][0pt]{$\m@th$=}%
}
% greater than or half-equal to
\newcommand{\gtheq}{\mathrel{\mathpalette\xgtheq\relax}}
\newcommand{\xgtheq}[2]{%
  \vcenter{\hbox{%
    \oalign{$\m@th#1>$\cr\lhalfeq{#1}\hidewidth\cr}%
  }}
}
% less than or half-equal to
\newcommand{\ltheq}{\mathrel{\mathpalette\xltheq\relax}}
\newcommand{\xltheq}[2]{%
  \vcenter{\hbox{%
    \oalign{$\m@th#1<$\cr\hidewidth\rhalfeq{#1}\cr}%
  }}
}
\makeatother

\begin{document}
$a\gtheq b\ltheq c$
\end{document}

Answer 2

Answer to problem #1:

Is this what you asked for? :

$\left\lfloor abc \right\rfloor \left\lceil xyz \right\rceil$

Answer 3

Thank you for your answers; egreg's answer was especially useful. I was unable to install package trimclip, though, which appears not to be part common TeX distributions, and thus I am afraid may constitute a problem down the road, when/if the paper is accepted for publication. So, I've tried to use egreg's ideas to do without a specialized package, and have come up with "half-equal" inequalities which look similar to egreg's, but of course "my solution" is too crude and ad hoc, and it doesn't scale; see the code and results below. Can this "solution" be improved, again without using a rare latex package? Perhaps I should have said where such symbols can be useful -- of course, it's in Fourier analysis of (possibly signed or even complex-valued) measures, where it is natural to define the distribution function of such a measure at every point of its discontinuity to be exactly in the middle between the left and right limits of the distribution function at such a point.

As for the the corresponding "half-equal" delimiters, I haven't been able to get a combination of a half-bracket and half-parenthesis as I would like to, but rather a combination of \lfloor and \rfloor, with which I guess I could live for a while.

Here are "my" code and the corresponding images (sorry, I don't know how to make the images larger):

\documentclass{article}  

\usepackage%[dvips]  
{color}  

%left half of =  
\newcommand{\lhalfeq}[1]{\mbox{$#1=$\kern-3.5pt{\textcolor{white}{\rule[1pt]{4pt}{3.4pt}}} } }  

%right half of =  
\newcommand{\rhalfeq}[1]{\mbox{$#1=$\kern-7.9pt{\textcolor{white}{\rule[1pt]{4pt}{3.4pt}}} } }  

% greater than or half-equal to    
\newcommand{\gtheq}{\mathrel{\mathpalette\xgtheq\relax}}  
\newcommand{\xgtheq}[2]{%  
  \vcenter{\hbox{%  
    \oalign{$#1>$\cr\lhalfeq{#1}\hidewidth\cr}%  
  }}  
}  
% less than or half-equal to  
\newcommand{\ltheq}{\mathrel{\mathpalette\xltheq\relax}}  
\newcommand{\xltheq}[2]{%  
  \vcenter{\hbox{%  
    \oalign{$#1<$\cr%\hidewidth  
    \rhalfeq{#1}\cr}%  
  }}\kern-3pt  
}

\begin{document}  

$(-\infty,b\rfloor\kern-3.9pt\lfloor$ \qquad  
%  
$\rfloor\kern-3.9pt\lfloor a,\infty)$ \qquad  
%  
$a\gtheq b\ltheq c$  

\end{document}